Dynamics of a Mosquito Egg-Larvae Model with Seasonality.

We propose a two stages mosquito egg-larvae model with seasonality as a simplification of a four stages one. For the simplified model we characterize the dynamics in terms of the vectorial reproduction number, [Formula: see text], obtaining extinction if [Formula: see text] and convergence to a unique positive periodic orbit if [Formula: see text]. We illustrate each case with an example, by providing general conditions on the periodic coefficients for its occurrence. These examples are further developed using numerical simulations where the periodic parameters satisfy the conditions obtained. In the [Formula: see text] case, real climatic data is used for inferring the parameter behaviour. For the four stage system, using alternative oviposition rate functions, we present a result which generalizes others given for models with delays and even with diffusion to the case in which competition between the larvae is introduced. The analytical study of our initial four stages system when [Formula: see text] remains open, since we were not able to prove that in this case the system is dissipative.

Fixed points for planar maps with multiple twists, with application to nonlinear equations with indefinite weight.

In this paper, we investigate the dynamical properties associated with planar maps which can be represented as a composition of twist maps together with expansive-contractive homeomorphisms. The class of maps we consider present some common features both with those arising in the context of the Poincaré-Birkhoff theorem and those studied in the theory of topological horseshoes. In our main theorems, we show that the multiplicity results of fixed points and periodic points typical of the Poincaré-Birkhoff theorem can be recovered and improved in our setting. In particular, we can avoid assuming area-preserving conditions and we also obtain higher multiplicity results in the case of multiple twists. Applications are given to periodic solutions for planar systems of non-autonomous ODEs with sign-indefinite weights, including the non-Hamiltonian case. The presence of complex dynamics is also discussed. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

On the correlation between variance in individual susceptibilities and infection prevalence in populations.

The hypothesis that infection prevalence in a population correlates negatively with variance in the susceptibility of its individuals has support from experimental, field, and theoretical studies. However, its generality has never been formally demonstrated. Here we formulate an endemic SIS model with individual susceptibility distributed according to a discrete or continuous probability function to assess the generality of such hypothesis. We introduce an ordering among susceptibility distributions with the same mean, analogous to that considered in Katriel (J Math Biol 65:237-262, 2012) to order the attack rates in an epidemic SIR model with heterogeneity. It turns out that if one distribution dominates another in this order then it has greater variance and corresponds to a lower infection prevalence for R0 varying in a suitable maximal interval of the form ]1, R0*]. We show that in both the discrete and continuous frameworks R0* can be finite, so that the expected correlation among variance and prevalence does not always hold. For discrete distributions this fact is demonstrated analytically, and the proof introduces a constructive procedure to find ordered pairs for which R0* is arbitrarily close to 1. For continuous distributions our conclusion is based on numerical studies with the beta distribution. Finally, we present explicit partial orderings among discrete susceptibility distributions and among symmetric beta distributions which guarantee that R0* = +∞.

Persistence in seasonally forced epidemiological models.

In this paper we address the persistence of a class of seasonally forced epidemiological models. We use an abstract theorem about persistence by Fonda. Five different examples of application are given.

Heterogeneity in susceptibility to infection can explain high reinfection rates.

Heterogeneity in susceptibility and infectivity is inherent to infectious disease transmission in nature. Here we are concerned with the formulation of mathematical models that capture the essence of heterogeneity while keeping a simple structure suitable of analytical treatment. We explore the consequences of host heterogeneity in the susceptibility to infection for epidemiological models for which immunity conferred by infection is partially protective, known as susceptible-infected-recovered-infected (SIRI) models. We analyze the impact of heterogeneity on disease prevalence and contrast the susceptibility profiles of the subpopulations at risk for primary infection and reinfection. We present a systematic study in the case of two frailty groups. We predict that the average rate of reinfection may be higher than the average rate of primary infection, which may seem paradoxical given that primary infection induces life-long partial protection. Infection generates a selection mechanism whereby fit individuals remain in S and frail individuals are transferred to R. If this effect is strong enough we have a scenario where, on average, the rate of reinfection is higher than the rate of primary infection even though each individual has a risk reduction following primary infection. This mechanism may explain high rates of tuberculosis reinfection recently reported. Finally, the enhanced benefits of vaccination strategies that target the high-risk groups are quantified.